1A person's mood, temper, or spirits—how they are feeling emotionally at a given time.
今日は部長の機嫌が悪くて、会議が早く終わった。
The manager was in a bad mood today, so the meeting ended early.
子供の機嫌を取るためにお菓子を買った。
I bought sweets to cheer up the child.
彼は機嫌によって態度が変わることがある。
His attitude sometimes changes depending on his mood.
2An older, more formal usage referring to someone's health, safety, or general condition—as in the set phrase ご機嫌伺い (inquiring after someone's well-being).
久しぶりに恩師にご機嫌伺いの手紙を書いた。
I wrote a letter to my old teacher, inquiring after their well-being, for the first time in a while.
祖父の機嫌を伺うため、毎週電話をかけている。
I call every week to check on my grandfather's health and well-being.
3Used (often as ご機嫌) to mean being in a good mood or high spirits; a polite/exclamatory way to ask or remark about someone's state.
祖母はご機嫌で昔話をたくさん話してくれた。
Grandma was in high spirits and told us many old stories.
ご機嫌いかがですか？
How are you? (a polite greeting to a superior or guest)
彼女はご機嫌な顔で歌を歌っている。
She is singing with a cheerful expression.